Solution: The equality \(\left|a+b\right| = \left|a\right| + \left|b\right|\) holds when:
1. Vectors in the Same Direction:
- If \( a \) and \( b \) are vectors, then \( |a+b|=|a|+|b| \) if and only if \( a \) and \( b \) point in the same direction or are scalar multiples of each other. This is because the triangle inequality becomes an equality when the vectors are collinear or lie along the same line.
2. Real Numbers \( a \) and \( b \) :
- For real numbers \( a \) and \( b \), the condition \( |a+b|=|a|+|b| \) holds if both numbers have the same sign or if one of them is zero.
- Case 1: Both Positive
- If \( a \geq 0 \) and \( b \geq 0 \), then \( |a+b|=a+b=|a|+|b| \).
- Case 2: Both Negative
- If \( a \leq 0 \) and \( b \leq 0 \), then \( |a+b|=-a-b=|a|+|b| \).
- Case 3: One is Zero
- If \( a=0 \) or \( b=0 \), the equality holds trivially as \( |a+b|=|a|+|b| \).
